12736

Searching for a counterexample of Kurepa’s Conjecture

Vladica Andrejic, Milos Tatarevic
Faculty of Mathematics, University of Belgrade, Belgrade, Serbia
arXiv:1409.0800 [math.NT], (2 Sep 2014)

@article{2014arXiv1409.0800A,

   author={Andreji{‘c}}, V. and {Tatarevic}, M.},

   title={"{Searching for a counterexample of Kurepa’s Conjecture}"},

   journal={ArXiv e-prints},

   archivePrefix={"arXiv"},

   eprint={1409.0800},

   primaryClass={"math.NT"},

   keywords={Mathematics – Number Theory, 11B83},

   year={2014},

   month={sep},

   adsurl={http://adsabs.harvard.edu/abs/2014arXiv1409.0800A},

   adsnote={Provided by the SAO/NASA Astrophysics Data System}

}

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Kurepa’s conjecture states that there is no odd prime p which divides !p=0!+1!+…+(p-1)!. We search for a counterexample of this conjecture for all p<10^10. We introduce new optimization techniques and perform the computation using graphics processing units (GPUs). Additionally, we consider the generalized Kurepa’s left factorial given as !kn=(0!)k+(1!)k+…+((n-1)!)k and show that for all integers 1<k<100 there exists an odd prime p such that p|!^kp.
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